Articles of complex analysis

Intuition for $e^{ix}$ lies on the unit circle.

I do know that for any $x$, the complex number $e^{ix}$ have to lie on the unit circle because if we plot the points for $\cos x + i\sin x$ for each $x$, we will eventually form a circle. Are there any more intuitive approach on why $e^{ix}$ lies on the unit circle?

Cauchy's Theorem vs. Fundamental Theorem of Contour Integration.

The fundamental theorem of contour integration says if one has a function and its antiderivative, and integrates the function over a closed loop the result is zero. Cauchy’s theorem (Goursat’s Version) says the integral of a function in a holomorphic domain in a closed loop is zero. Cauchy’s theorem is apparently much stronger, the proof […]

The sum of the residues of a meromorphic differential form on a compact Riemann surface is zero

How can one see that the sum of the residues of a meromorphic function on a Riemann surface $ \Sigma_g$ of positive genus is always zero? This is not true for the Riemann sphere $\mathbb{CP}^1$.

Why is an analytic variety irreducible provided that the set of its smooth points is connected?

Here is a proposition from Griffith and Harris, Principles of Algebraic Geometry. Let $V$ be an analytic variety on a complex manifold $M$, let $V^*$ be its points of smooth points. If $V^*$ is connected, then $V$ is irreducible. The proof in the book is very short. It goes like this: Suppose $V=V_1\cup V_2$ is […]

An entire function which is real on the real axis and map upper half plane to upper half plane

Suppose that $f$ is an entire function that satisfies $f(z)$ is real when $z$ is real and if $Imz>0$ then $Imf(z)>0$. Prove that $f$ can have at most one zero and that the zero, if it occurs, is real. Show also that if $f$ has no zero then $f$ is constant. By the open mapping […]

A question regarding Frobenius method in ODE

Suppose $b(x),c(x)$ are real functions analytic at $0$. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. Suppose $r$ is a double root of $r(r-1)+b_0r+c_0=0$. It is well known that the differential equation $$x^2y”+xb(x)y’+c(x)y=0$$ has a solution of the form $$y_1=x^r(1+\sum_{i=1}^\infty a_ix^i),$$ where the series $\sum_{i=1}^\infty a_ix^i$ has radius of convergence $\ge R$ (e.g., see Tyn Myint-U, […]

Convergence of the taylor series of a branch of logarithm.

Consider the branch of log defined on $\mathbb{C}$ with the negative real axis and origin removed. I was told that its Taylor series about the point $z_0 = -2 + i$ converges in a radius $\sqrt5$, which means the Taylor series actually converges for points on the negative real axis (not in its domain). I […]

How to show $e^{2 \pi i \theta}$ is not algebraic.

I was wondering if someone could possibly help me figure out how to show $e^{2 \pi i \theta}$ is not algebraic when $\theta$ is irrational. Thanks!

Closed form for an infinite series of Bessel functions

I have come across an infinite series involving Bessel functions where the summation is over the argument inside the Bessel function (rather than over the index of the Bessel function, which seems to be the case usually studied). Specifically I am wondering whether a closed form is known for $$\sum_{n=1}^{\infty} \frac{J_k(nz)}{n^k}.$$ Here, $k$ is an […]

Roots of a polynomial

I am working the next problem: Consider the polynomials $$ p_n(z)=\sum_{j=0}^{n}\frac{z^j}{j!} $$ For $n \geq 2$, show that if $a \in \mathbb{C}$ is such that $|a|=1$ or $|a|=n$, then $p_n(a)\neq 0$ For the case $|a|=1$, I think i have a partial solution: Suppose that $a\in \mathbb{C}$ is such that $p_n(a)=0$ and $|a|=1$, then by the […]